begin{subfigure}[0.45textwidth]
    includegraphics[width=textwidth]{long_pneumonic.jpg}
    caption{Pre-vaccination susceptible and infected
    population$alpha=0.000015, beta=0.0027, delta=.002, sigma=1/3,
    Delta=3/4, r=1/5$.}
end{subfigure}
begin{subfigure}{0.45textwidth}
    includegraphics[width=linewidth]{long_pneumonic2.jpg}
    caption{Transient loops of the infected and susceptible population
    changes.}
end{subfigure}
begin{subfigure}{0.45textwidth}
    includegraphics[width=linewidth]{long_pneumonic3.jpg}
    caption{The population of susceptible individuals and infected
    individuals over a period of 1500 days post-vaccination.
    $alpha=0.000015, beta=0.0027, delta=.002, sigma=1/3, Delta=3/4,
    r=1/5$.}
end{subfigure}
begin{subfigure}{0.45textwidth}
    includegraphics[width=linewidth]{long_pneumonic4.jpg}
    caption{[A graph showing the rise and decrease of pneumonic plague
    victims leading to increasing amounts of infected with each recurrence
    post-vaccination.}
end{subfigure}
caption{Graphs comparing and contrasting the same parameters examining the
dynamics of the susceptible and infected populations pre and post
vaccinations.}